Scattered space

Scattered spaces have their origin in Cantors investigations into the uniqueness of trigonometric series. His theorem reads (in modern terminology): If the partial sums of a trigonometric series

$$ \frac{a _ {0} }{2} + \sum _ {n= 1 } ^ \infty ( a _ {n} \cos nx+ b _ {n} \sin nx) $$

converge to zero except possibly on a set of points of finite scattered height, then all coefficients of the series must be zero.

Scattered spaces and their scattered height are defined as follows. One first defines for any space $ X $ a transfinite sequence $ \langle X ^ {( \alpha ) } \rangle _ \alpha $ of subspaces: Let $ X ^ {( 0) } = X $, for any ordinal $ \alpha $, let $ X ^ {( \alpha + 1) } $ be the derived set of $ X ^ {( \alpha ) } $, and if $ \lambda $ is a limit ordinal, put $ X ^ {( \lambda ) } = \cap _ {\alpha < \lambda } X ^ {( \alpha ) } $.

There is a first ordinal $ \alpha = \alpha _ {X} $ for which $ X ^ {( \alpha ) } = X ^ {( \alpha + 1) } $. If this derived set $ X ^ {( \alpha ) } $ is empty, one calls $ X $ a scattered space and the ordinal $ \alpha _ {X} $ its scattered height.

If $ X $ is compact, then it is readily seen that $ \alpha _ {X} $ must be a successor ordinal, say $ \alpha _ {X} = \beta + 1 $. The set $ X ^ {( \beta ) } $ is finite. It is a classical result of S. Mazurkiewicz and J. Sierpiński that a countable compact scattered space $ X $ is determined completely by the ordinal $ \beta $ and the number $ n $ of points in $ X ^ {( \beta ) } $: $ X $ is homeomorphic to the set of ordinal numbers less than or equal to $ \omega ^ \beta \cdot n $.

Compact scattered spaces correspond, via Stone duality, to the so-called superatomic Boolean algebras; these are defined to be those algebras for which every (non-trivial) homomorphic image has an atom. Because of this duality one may say that the structure of compact scattered spaces is understood best.

An important family of scattered spaces (of height $ 2 $) is constructed as follows: Take an infinite set $ X $ and a family $ {\mathcal A} $ of countably-infinite subsets that is almost disjoint, i.e., if $ A,B \in {\mathcal A} $, then $ A \cap B $ is finite. The union $ X \cup {\mathcal A} $ is topologized by declaring $ X $ to be an open discrete subspace and giving an element $ A $ of $ {\mathcal A} $ basic neighbourhoods of the form $ \{ A \} \cup A\setminus F $, where $ F $ is a finite subset of $ X $. By varying the family $ {\mathcal A} $ one can obtain various interesting examples of topological spaces, for example, in this way one can make a pseudo-compact space that is not countably compact (cf. Countably-compact space).

References

[a1]	G. Cantor, "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen" Math. Ann. , 5 (1872) pp. 123–132 Zbl 04.0101.02
[a2]	S. Mŕowka, "On completely regular spaces" Fund. Math. , 41 (1954) pp. 105–106 Zbl 0055.41304
[a3]	J. Roitman, "Superatomic Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1–3 , North-Holland (1989) pp. Chapt. 19; pp. 719–740 Zbl 0671.06001